The Bohr-van Leeuwen theorem and the thermal Casimir effect for conductors

The problem of estimating the thermal corrections to the Casimir and Casimir-Polder interactions in systems involving conducting plates has attracted considerable attention in the recent literature on dispersion forces. Alternative theoretical models, based on distinct low-frequency extrapolations of the plate’s reflection coefficient for transverse electric TE modes, provide widely different predictions for the magnitude of this correction. In this paper we examine the most widely used prescriptions for this reflection coefficient from the point of view of their consistency with the Bohr–van Leeuwen theorem of classical statistical physics, stating that at thermal equilibrium transverse electromagnetic fields decouple from matter in the classical limit. We find that the theorem is satisfied if and only if the TE reflection coefficient vanishes at zero frequency in the classical limit. This criterion appears to rule out some of the models that have been considered recently for describing the thermal correction to the Casimir pressure with nonmagnetic metallic plates.